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B. Srinivasa Rao, M. Rami Reddy and K. Rosaiah RT&A, No 3 (74)
ATTRIBUTE CONTROL CHART FOR TLT USING EIKD Volume 18, September 2023
AN ATTRIBUTE CONTROL CHART FOR TIME TRUNCATED LIFE TESTS USING EXPONENTIATED INVERSE KUMARASWAMY DISTRIBUTION
B. Srinivasa Rao1, M. Rami Reddy2 and K. Rosaiah3
department of Mathematics & Humanities, R.V.R& J.C College of Engineering, Chowdavaram, Guntur- 522 019, A.P, India 2Freshman Engineering Department, Lakireddy Bali Reddy College of Engineering,
Mylavaram-521 230, A.P, India 3Department of Statistics, Acharya Nagarjuna University, Guntur, A.P, India3
[email protected], [email protected], [email protected]
Abstract
In this article an attribute control chart is designed for the Exponentiated Inverse kumaraswamy distribution under a time truncated life test by assuming the life-time of the item follow the selected Exponentiated Inverse Kumaraswamy distribution with known parameters. In order to limit the cost of checking the quality of an item in any industrial process with time truncation, this process is much useful. By considering the average number of defective items from a specified lot that are failed before the time limit, the attribute control limits are constructed. The control chart is determined using Binomial distribution based on the Upper and Lower control limits. The functioning of the designed control chart is examined with the average run length (ARL) values. The control chart constants and limits are calculated for specific ARL values with assumed parameters at different sample sizes for an in-control process. These control chart constants are obtained by considering different combinations of parameters of the assumed distribution. With these in-control limits the ARL values are observed by shifting the parameter values. A simulation analysis is developed by taking a specific number of observations in each sample and the average number of failures from each sample is considered as a statistic to establish the execution of the control chart for a specified ARL at a particular shift in parameter. With that statistic of average number of failures from the samples the control chart is prepared. It is observed a specific change in defective number when there is shift in parameter values. The results are illustrated with an example.
Keywords: Exponentiated Inverse Kumaraswamy distribution; attribute control chart; time truncated life test; average run length; simulation.
1. Introduction
To examine the quality of an article in industrial production, control charts are much helpful. Simultaneously it is very important to maintain the standards or even to improve the quality of article to meet customer satisfaction levels. It requires a regular monitoring to assess the quality of the articles. To face the competition in the market it is very important to complete this screening process with less cost and within a shortest possible time. It is a common practice that the process is considered to be in control when the examined statistic values lies within the control limits known as Upper and Lower control limits (UCL and LCL) that are also believed as the extremities for specification of the product's quality. If the sample points exceeding the limits then the production procedure is treated as intemperate and these items are considered to be defective or
B. Srinivasa Rao, M. Rami Reddy and K. Rosaiah RT&A, No 3 (74) ATTRIBUTE CONTROL CHART FOR TLT USING EIKD_Volume 18, September 2023
imperfect. In terms of reducing the defective items or refining the quality of the items in a less span
of time and with minimal cost, the control chart methods are much beneficial.
In general we have two kinds of control charts for variables and attributes. Variable control charts can be applied to any quality characteristic that is measurable. Whereas attribute control charts are useful in classifying defective and non-defective items in the production process. There are several studies designed by various authors that the construction and implementation of different attribute and variable control charts. A few of them are Epprecht et al.[1] studied about the Adaptive control charts for attributes. Wu et al. [2] prepared an optimal np chart with curtailment. Ho and Quinino [3] discussed the monitoring process of variability through an attribute control chart. Wu and Wang [4] proposed np-control chart using double inspection. Further, some more attribute and variable controlcharts also established in Chiu and Kuo [5], A.D. Rodrigues et al. [6], Joekes and Barbosa [7], Arif et al. [8], and Shafqat et al. [9].
Generally most of the constructing processes of control charts are based on the supposition that the quality of items follow normality. While some circumstances where the its characteristic is unknown or doesn't follow the normal distribution. Various authors developed the procedure for the construction of control charts for non normal distributions, for example: Bai and choi [10], Chang and Bai, [11], Al-Oraini and Rahim [12], Aslam et al. [13] and Lin and Chou [14].
As it is essential for any industry to sustain in the competitive market by manufacturing more reliable products with less cost, with less manpower in a short period time it requires less time for inspection of defective products. To achieve this, it is necessary to have a time truncated life test based control chart. Hence, preparing a control chart for monitoring a non-normality characteristic product under the time truncated test is preferred to inspect the lifetime of the product.
Various authors proposed articles to expand the methodology of constructing control charts for various distributions under time truncated life test. A few references of such models are Aslam and Jun [15] developed a time truncated life test(TLT) based attribute control chart for Weibull distribution. Aslam et al. [16] designed a TLT based control chart for Pareto distribution of second kind. Similarly, Rao [17] proposed for exponentiated half-logistic distribution. Rosaiah et al. [18] considered for exponentiated Frechet distribution. Shruthi. G and O.S.Deepa [19] monitored ARL for Exponentiated distributions under TLT. Rao et al. [20] introduced TLT based chart for Dagum distribution. Adeoti and Ogundipe [21] developed for generalized exponential distribution under TLT. Rosaiah et al. [22] designed for type-II generalized log logistic distribution. Jafarian-Namin et al. [23] studied an efficient design of attribute control chart under TLT for weibull distribution. G. S .Rao and Al-Omari [24] designed for Length-Biased Weighted Lomax Distribution. Baklizi and Ghannam [25] proposed a TLT based attribute control chart for the inverse Weibull distribution. Our interest is to develop an article of TLT based attribute control chart for monitoring the quality process when the lifetime of an article follow Exponentiated Inverse kumaraswamy distribution (EIKD). To monitor the functioning of a control chart ARL is general procedure which gives the
average number of values that must be considered before an observation signals as out-of control.
i
The ARL is determined asARL = -; here P is the probability of any observation indicates out of control. The article is summarized in the following way: a concise introduction of the Exponentiated Inverse kumaraswamy distribution (EIKD) is provided in section-2, design and execution of control chart with calculations of ARL values when the parameter is shifted for EIKD is discussed with an application in Section-3. The process is evaluated with an analysis of simulation data in section-4. Few closing remarks are specified in Section-5.
2. Exponentiated Inverse Kumaraswamy distribution
The probability density function (pdf) of the Exponentiated inverse Kumaraswamy distribution (EIKD) is
f(x) = a$X( 1 + x)-(a+1)(1-(1 + x)-a)^x-1 ;0<x<™, a,(3,X>0 (1)
Its cumulative distribution function (cdf) is given by,
F(x) = [1 - (1 + x)-a]Px ; 0 < x < œ, a,fi,X> 0 (2)
Where a, p and X are shape parameters The mean of EIKD isE(X) = Xfi B (l -1 , fiX), a > 1 The Reliability function is
R(x) = 1 - [1 - (1 + x)-a]px (3)
and the Hazard function is
„, . aXp( 1+x)-(a+1)[1-(1+x)-a]^i-1 „ ^ n
H(x) = —---—;—r1—ri- ,0 < x < œ and X,a,B > 0 (4)
v J 1-[1-(1+x)-a]W r w
Graphs of the pdf, cdf, Reliability and hazard functions of EIKD for selected parameter values are plotted respectively.
0,6 0,5 0,4 f(x) 0,3
0,2 0,1 0
0
a=2.5, ß=2.25, Л=0.5 a=2.5, ß=2.25, Л=1.0 a=2.5, ß=2.25, Л=1.5 a=2.5, ß=2.25, Л=2.0 a=2.5, ß=2.25, Л=2.5
F(x)
1
0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0
0123456789 10
Figure 1: The pdf plots of EIKD
Figure 2: The cdf plots of EIKD
R(X)
1
0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0
a=2.5, ß=2.25, Л=0.5 a=2.5, ß=2.25, Л=1.0 a=2.5, ß=2.25, Л=1.5 a=2.5, ß=2.25, Л=2.0 a=2.5, ß=2.25, Л=2.5
0123456789 10
H(X)
1,4 1,2 1 0,8 0,6 0,4 0,2 0
0
a=2.5, ß=2.25, Л=0.5 a=2.5, ß=2.25, Л=1.0 a=2.5, ß=2.25, Л=1.5 a=2.5, ß=2.25, Л=2.0 a=2.5, ß=2.25, Л=2.5
46
X
10
12
Figure 3: Reliability function plots of EIKD
Figure 4: Hazard function plots of EIKD
2
6
8
X
2
8
X
3. Designing of the control chart
To construct the 'np' chart on the basis of defective articles in the production process the following methodology was implemented
Stepl: A sample of 'n' articles is considered randomly from every subgroup lot and apply time truncated life test for these articles. Consider the number of articles (D) that are out of order (failure) within the termination timet0stated as t0 = a^0, here ais constant related with target average life n0 when the process is supposed to be in-control.
Step2: Declare the process is under control if D rests in the limits of LCL and UCL, else, if D>UCL or D<LCL it is declared as not in control. Since D is a distinct count out of a sample of 'n' items, it can be considered as a "Binomial variate" asn and p are parameters the control limits for in-control process are given as;
(5a) (5b)
. UCL = npo + k^npo(l - Po)
. LCL = MAX[0, np0 - k^np0(T—p00)]
Here'P0'is the probability of an article is bungled earlier thant0and it is determined from equation-(2) as"p0 = F(t0)"and k is the constant of the control chart. However, we state the process is in-control when^ = (or the parameters a = a0,p = p0 and A = A0). Thenp0 is obtained from equation (2) as
P0 = F[t0; a0,P0,A0] = [1-(1 + t0)- a°]
loh.Bi}-!, №)} (6)
In real time applications, the probabilityp0 is typically unknown; then the limits for such situations are
1-\l + a
UCL = D + k
LCL = Max
0, D-k
Here£>is mean count of failure articles in the samples. The probability P?j of confirming in control process of the planned chart is specified as
p0 _ yUCL
rin = Ld=LCL+1
Q(P0)d(1-P0)
n—d
(7a) (7b)
(8)
P°n = P[LCL <D < UCL/P0]-
UCL
fW
d=LCL+l
I ©ji1-^^^1--,^)
—a0
1 +
, 1 y — «°
ap0A0.B(1-—,p0A0) \ a0 /
,n—d
(9)
The control charts efficacy can be examined by its "Average Run Length (ARL)", a nd is defined when the process is under control as
ARU =
0 1—p° in
(10)
3.1. ARLs with a shift in Parameter
To examine the performance of the control chart with a shift in one of the parameter (X) as A1 = cA0, here c is specified as shift constant
The probability of an article is out of order prior to the experimental time t0 is consider asp1; and is obtained as
Pi = F[t0; ao,po,t.1] = [1 -(1 + t0)— a°] ^
d
1
1-{l + a ß0b.B(l-± ß0Ao]}
■ ao
ß0cÄ0
The in-control probability of the process with the parameter shift as
pi = У
1 in ¿Ji
pin = P[LCL <D< UCL/P1]=
UCL
d=LCL+1 ( ^
Q(Pi)da-Pi)
n—d
1 G);
d=LCL+1
1 -
1 + aß0Ä0.B(l-—,ß0Ä0) \ a0 )
— a0
ßo^l
1 + aß0Ä0. В (1
0.B(l-—,ß0Ä0)
\ a0 j
~<*0
ßo^i
The ARL for the process shift is given as
1
ARL1 = —-71 i—PL
(11)
(12)
j«—d
(13)
(14)
The approach for the calculations of intended chart is mentioned below
(1) Choose the ARL (sayr0), parameter values (a0, p0 and X0), and the constant a.
(2) Control chart parameters to be determined for a specified sample size n, provided^fiL0 which is
specified in Eq-(10) very near tor0that isARL0 > r0.
(3) The parameters obtained in previous step are utilized to calculate ARL1 as per the shift
constant c using Eq-(14).
The control limits of the chart are determined for various parameter values andr0 values that are shown in the Tables 1 through 8.We have noticed a rapid reducing tend in values with the decrement in shift value 'c'.
Table 1: ARL± Values for the designed chart with n=20; 20 = a0 = 2.5 and p0 = 2.5
d
L C L 2 1 1 2
U C L 14 13 14 15
a 0.3224 0.2891 0.3246 0.3512
k 2.8561 2.9154 2.9628 2.9965
ARLo=200 ARLo=250 ARLo=300 ARLo=370
C ARL, ARL( ARLj ARLj
1 200.088 250.258 300.372 370.897
0.9 118.620 120.348 119.549 178.751
0.8 49.315 46.023 46.680 71.215
0.7 20.178 18.085 19.162 28.765
0.6 8.803 7.716 8.452 12.287
0.5 4.212 3.674 4.090 5.664
0.4 2.273 2.014 2.231 2.888
0.3 1.434 1.319 1.420 1.686
0.2 1.095 1.058 1.091 1.178
0.1 1.005 1.002 1.004 1.014
Table 2: ARL^Values for the designed chart with n=20; 20 = l.B,a0 = 2 and p0 = 2.5
LCL 5 1 0 1
UCL 17 13 12 14
a 0.4858 0.3015 0.2724 0.3328
k 2.8126 2.8682 2.9864 2.9962
ARLo=200 ARLo=250 ARLo=300 ARLo=370
C ARLl ARLl ARLl ARLl
1 200.071 250.140 300.364 370.482
0.9 162.945 120.234 111.599 149.679
0.8 84.533 45.983 40.166 56.697
0.7 39.198 18.072 15.478 22.461
0.6 18.309 7.711 6.580 9.563
0.5 8.865 3.672 3.172 4.471
0.4 4.520 2.014 1.792 2.362
0.3 2.479 1.319 1.229 1.462
0.2 1.514 1.058 1.034 1.101
0.1 1.094 1.002 1.001 1.005
Table 3: ARL1Values for the designed chart with n=20; 20 = 2, a0 = 2 and ß0 = 3
LCL 5 1 1 0
UCL 17 13 14 13
a 0.5421 0.351 0.3909 0.3498
k 2.6423 2.8645 2.9641 3.0125
ARLo=200 ARLo=250 ARLo=300 ARLo=370
C ARLX ARLl ARLX ARLX
1 200.555 250.200 300.514 370.295
0.9 161.470 120.292 119.607 131.616
0.8 83.529 46.003 46.699 47.893
0.7 38.777 18.079 19.169 18.611
0.6 18.146 7.714 8.455 7.885
0.5 8.803 3.673 4.091 3.730
0.4 4.497 2.014 2.231 2.033
0.3 2.471 1.319 1.420 1.325
0.2 1.512 1.058 1.091 1.059
0.1 1.094 1.002 1.004 1.002
Table 4: ARL± Values for the designed chart with n=20; 20 = 5, a0 = 2.5 and p0 = 3
LCL 2 0 1 0
UCL 14 12 14 12
a 0.4198 0.3598 0.4221 0.3488
k 2.7562 2.8163 2.9564 2.9688
ARLo=200 ARLo=250 ARLo=300 ARLo=370
C ARL, ARLl ARL, ARLl
1 200.081 250.221 300.745 370.192
0.9 118.599 92.574 119.700 141.897
0.8 49.306 34.284 46.731 49.380
0.7 20.175 13.640 19.179 18.264
0.6 8.802 5.984 8.458 7.455
0.5 4.211 2.972 4.092 3.456
0.4 2.273 1.726 2.231 1.885
0.3 1.434 1.210 1.420 1.256
0.2 1.095 1.031 1.091 1.039
0.1 1.005 1.001 1.004 1.001
Table 5: ARL± Values for the designed chart with n=30; 20 = 1. 5, a0 = 2.5 and p0 = 2.25
LCL 9 3 6 8
UCL 24 18 22 24
a 0.4381 0.2897 0.2685 0.4246
k 2.7864 2.8636 2.9238 2.9817
ARLo=200 ARLo=250 ARLo=300 ARLo=370
C ARL! ARL, ARL! ARL,
1 200.112 250.372 301.927 370.298
0.9 166.463 90.656 122.171 270.239
0.8 69.870 28.891 43.166 101.717
0.7 26.828 10.262 16.265 36.301
0.6 10.954 4.257 6.789 13.868
0.5 4.923 2.138 3.227 5.846
0.4 2.509 1.344 1.806 2.805
0.3 1.507 1.067 1.232 1.597
0.2 1.111 1.004 1.034 1.132
0.1 1.006 1.000 1.001 1.007
Table 6: ARL^Values for the designed chart with n=30; 20 = 1. 5, a0 = 2 and p0 = 2.5
LCL 9 3 2 4
UCL 24 18 17 20
a 0.4657 0.3021 0.27939 0.3409
k 2.7951 2.8783 2.9438 2.9857
ARLo=200 ARLo=250 ARLo=300 ARLo=370
C ARL! ARL! ARL, ARL!
1 200.157 250.489 300.090 370.029
0.9 166.398 90.720 96.226 136.724
0.8 69.836 28.908 29.187 43.416
0.7 26.817 10.267 10.039 15.051
0.6 10.951 4.258 4.087 5.950
0.5 4.922 2.138 2.045 2.774
0.4 2.509 1.344 1.301 1.584
0.3 1.507 1.067 1.053 1.141
0.2 1.111 1.004 1.003 1.014
0.1 1.006 1.000 1.000 1.000
Table 7: ARL1Values for the designed chart with n=30; A,0 = 2, a0 = 2 and p0 = 3
LCL 9 2 1 5
UCL 24 17 16 21
a 0.5207 0.3327 0.31 0.415
k 2.7642 2.8645 2.9368 2.9864
ARLo=200 ARLo=250 ARLo=300 ARLo=370
C ARL, ARL, ARL, ARL,
1 200.133 250.941 300.492 370.132
0.9 166.433 78.483 84.867 165.081
0.8 69.854 24.700 25.461 53.743
0.7 26.823 8.846 8.813 18.549
0.6 10.952 3.744 3.651 7.198
0.5 4.922 1.941 1.881 3.245
0.4 2.509 1.271 1.241 1.766
0.3 1.507 1.047 1.038 1.202
0.2 1.111 1.002 1.001 1.025
0.1 1.006 1.000 1.000 1.000
Table 8: ARL^Values for the designed chart with n=30; 20 = 2. 5, a0 = 2. 5 and p0 = 3
LCL 9 3 4 8
UCL 24 18 20 24
a 0.5421 0.3848 0.43039 0.5279
k 2.7123 2.8215 2.9452 2.9938
ARLo=200 ARLo=250 ARLo=300 ARLo=370
C ARL, ARL, ARL, ARL,
1 200.041 250.287 300.702 370.128
0.9 166.565 90.609 103.844 270.558
0.8 69.926 28.878 34.402 101.855
0.7 26.846 10.259 12.545 36.343
0.6 10.960 4.256 5.211 13.880
0.5 4.925 2.137 2.545 5.849
0.4 2.510 1.344 1.513 2.806
0.3 1.507 1.067 1.122 1.598
0.2 1.111 1.004 1.012 1.132
0.1 1.006 1.000 1.000 1.007
3.2 Application of intended chart
To establish the applicability of the intended chart for the improvement of the quality of any manufactured article, we assume that the lifespan of the article follows the EIKD with parameters a0 = 2.5,p0 = 2.25 and A0 = 1.5. Consider the aimed mean life of the article is set as 1000 hrs with size of the sample n=20 of each group. If the target in control ARL value is fixed asr0 = 300for the control chart, as shown in Table 1, we obtain a= 0.3246, k=2.9628. From Equation-(6) we get p0 = 0.4221. Using Equations-(5a & 5b) the LCL and UCL are determined as LCL =1 and UCL = 14. Then, the functioning of the prepared control chart is as follows:
Step 1: Take a sample of 20 articles from each subgroup and test their lifespan for period of
324.6 hours. Figure out the failure count of articles (D) through the test. Step 2: We assert the process is in-control if 1 < D < 14, if not it is not in control.
4. Simulation study
A simulation study is presented to monitor the applicability of the designed control chart. It is executed using EIKD with specified parameters and the construction process is as follows:
The data is originated using EIKD with parameters ^o = 2, ^o = 2.5 and Ao = 1.5.A subgroup of 15 samples of size n=20 each are taken by considering the ARL asr0 = 300.The process is affirmed as in control with these parameter values when^0 = 3.5483. Another subgroup of 15 samples each of size n=20 are taken from EIKD with shift in parameter^ = cA0with shift value c =0.7. The control chart coefficient k=2.9864 is considered from Table 2, with ARL value as r0 = 300 and n=20 when the process is in control.
The termination time of the life-test will bet0 = a^0 = 0.2724 x 3.5483 = 0.9665. The number of articles that are failed before the termination timet0is considered as D which is calculated and presented in Table 9 for each sample. The number of average failures is£> = 6, the control limits are determined from equations (7a & 7b) are UCL=12 and LCL=0. The points of failure count (D) of each sample are exhibited in Figure 5. It is clearly observed that the planned chart indicated the shift at 18th sample (3rd sample after change in shift) while the corresponding
ARL value is 15. Thus the proposed chart effectively identifies the shift in this process.
5. Conclusion
In this article, we have designed a new ' np' control chart by assuming lifetime of the product follows EIKD to monitor the quality of manufactured articles under time truncation. The designed chart is then assessed by ARL's acquired from simulation study for different sample sizes; parameter values and objective in-control ARLs have been considered. The functioning of the intended chart is described with an explanatory example. For advance research, anyone can review applying the suggested control chart for any significant lifetime distribution. We have the feasibility to consider more quickened testing design to develop appropriate control charts for such circumstances.
Table 9: The simulation Analysis
1 2 3 4 5 6 7 8 9 sample 10 11 12 13 14 15 16 17 18 19 20 D
1 2.10 3.23 1.14 0.53 2.34 2.41 0.84 0.06 2.06 3.02 2.27 0.45 0.52 3.45 3.21 2.51 2.11 0.15 1.56 1.33 6
2 2.79 0.30 3.43 0.94 0.58 0.17 3.25 1.83 0.85 0.38 1.44 2.11 2.09 0.34 2.60 2.52 0.73 1.91 0.31 2.12 9
3 1.18 2.15 3.23 1.61 0.08 3.37 1.51 2.20 2.84 2.84 0.52 2.69 0.94 2.10 3.52 3.30 1.82 2.98 1.28 0.52 4
4 3.38 1.38 3.03 0.10 0.88 0.35 0.91 0.54 2.80 2.49 0.64 2.73 1.56 2.54 2.64 1.22 1.43 3.46 0.77 3.24 7
5 3.25 0.64 2.58 0.69 0.36 0.81 3.55 2.96 1.15 1.07 1.07 2.01 3.00 1.40 2.68 0.88 0.69 2.57 1.74 3.25 6
6 1.74 0.11 2.75 0.05 1.55 2.19 0.28 1.78 2.05 2.11 2.30 2.83 0.84 0.03 1.31 1.60 0.34 2.75 1.28 2.28 6
7 3.35 2.75 1.77 0.32 2.32 3.27 2.15 1.76 1.65 1.07 0.17 2.60 1.33 3.36 0.53 2.39 2.90 3.14 3.54 0.44 4
8 2.32 1.80 0.74 3.31 1.48 3.35 0.38 2.65 0.97 1.42 1.86 3.03 1.73 0.03 3.55 0.21 1.98 0.40 0.99 2.38 5
9 1.23 1.05 3.49 3.45 0.49 1.87 1.97 2.78 2.67 1.76 1.96 1.12 2.32 1.43 3.04 1.46 2.34 0.62 3.15 1.10 2
10 2.57 0.08 0.51 0.32 2.34 3.34 0.13 0.09 3.16 2.60 0.57 1.12 1.21 2.31 2.57 2.71 1.34 0.85 1.66 0.52 8
11 2.65 2.46 2.20 2.70 2.64 0.13 1.10 1.54 1.93 0.19 3.26 2.41 2.40 1.84 0.20 1.58 2.89 0.18 0.84 2.53 5
12 1.17 3.47 1.14 2.79 1.30 1.64 0.34 2.53 3.30 2.31 3.18 3.30 1.93 2.75 2.64 1.33 1.00 2.49 0.61 2.96 2
13 1.18 0.58 1.24 0.61 2.04 2.93 1.52 2.79 2.60 1.00 1.72 2.39 3.10 1.96 1.32 2.87 0.62 2.25 1.76 1.63 3
14 2.88 2.56 1.20 2.04 1.54 1.54 1.81 0.43 0.25 3.02 0.41 1.61 2.06 1.21 2.84 3.36 1.71 0.69 1.83 2.09 4
15 2.23 2.80 3.48 0.95 3.34 2.70 2.64 3.13 2.66 0.99 0.06 0.28 2.66 2.00 2.27 2.09 0.56 3.37 0.04 1.95 5
16 1.63 1.61 0.46 2.62 2.37 0.45 0.63 1.96 2.03 1.30 0.92 2.73 1.71 0.26 1.58 2.08 1.30 0.32 2.84 1.82 6
17 2.12 0.04 0.22 0.05 0.60 2.12 0.94 1.23 2.58 1.21 1.37 0.95 2.19 0.30 1.20 1.85 0.53 0.39 0.93 1.46 10
18 0.93 0.43 0.94 0.95 0.86 0.37 0.65 0.98 1.60 2.93 0.17 2.51 0.94 0.70 0.87 1.19 2.17 1.66 0.58 0.25 13
19 2.51 1.74 0.31 2.66 1.58 0.57 1.58 0.80 2.33 2.90 1.34 2.04 1.41 1.77 2.82 0.13 2.81 1.82 0.34 2.58 5
20 1.67 1.34 2.03 0.09 1.06 2.78 1.98 2.22 0.81 2.24 1.48 0.19 0.13 0.85 1.35 1.90 0.16 1.04 1.31 2.93 6
21 1.16 1.00 0.57 2.45 2.17 2.91 0.15 2.08 1.70 2.94 0.50 1.62 2.04 1.47 0.86 2.75 2.36 1.61 0.69 2.68 5
22 2.46 0.60 2.16 1.92 1.30 1.85 2.61 0.53 0.40 2.45 1.78 2.85 1.45 2.54 0.43 0.63 2.63 1.14 1.72 2.24 5
23 0.99 0.25 2.70 2.79 2.41 2.95 1.26 1.82 2.30 2.06 0.88 1.18 2.45 2.53 0.45 2.36 2.09 0.18 0.34 2.95 5
24 2.12 1.94 1.25 2.93 0.33 2.31 2.35 0.86 2.54 2.21 0.86 0.22 0.54 1.56 0.97 0.40 1.36 2.54 1.51 1.92 6
25 2.10 0.03 1.26 1.60 0.69 0.16 1.06 0.51 0.95 0.94 0.95 1.92 0.59 0.70 2.65 2.39 0.56 0.10 0.54 2.53 12
26 1.68 1.18 2.49 2.17 0.67 0.61 0.24 0.48 1.28 2.68 0.98 1.07 2.97 0.73 2.14 0.75 2.37 1.81 1.46 1.50 6
27 2.81 1.35 1.02 1.20 1.76 0.85 0.88 2.29 2.27 2.44 1.44 0.73 0.28 2.75 1.40 0.65 1.42 2.37 1.10 1.94 5
28 0.05 2.43 2.41 0.47 2.15 2.54 0.23 1.57 1.39 0.00 1.70 1.37 1.10 2.03 2.48 0.43 0.75 0.80 0.06 0.17 9
29 2.99 2.57 1.53 2.41 2.16 0.99 0.44 1.77 2.12 0.95 1.03 1.96 2.72 0.22 2.01 1.88 1.59 1.22 2.19 0.33 4
30 2.35 0.50 2.97 0.13 2.69 2.30 0.64 2.77 1.70 2.56 1.03 0.08 2.93 0.83 0.17 2.91 0.39 2.65 2.28 1.03 7
14
12
10
UCL=12
LCL=0
I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
sample number
8
6
4
2
0
Figure 5: Control chart for simulation data.
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